Continued Fractions and Hermite's Differerential Equation

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The continued fraction method for solving the differential equations of quantum mechanics is not well known, but since we will use the method to deal with tunneling problems and the Mathieu Equation, it makes sense to familiarize ourselves with the method, in a non-rigorous manner.

Hermite polynomials are usually generated using the differential equation and a power series Ansatz which requires truncation. An alternative approach is presented here. Hermite's differential equation is

$\begin{displaymath} \frac{\partial^2 y}{\partial x^2} - 2 x \frac{\partial y}{\partial x} +2 \alpha y=0 \end{displaymath}$

(1)

which we rewrite as

$\begin{displaymath}y''-2xy'+2\alpha y=0 \end{displaymath}$

using the ``prime'' notation scheme, i.e.,

$\begin{displaymath}y' = \frac{\partial y}{\partial x} \end{displaymath}$

$\begin{displaymath}y'' = \frac{\partial^2 y}{\partial x^K} \end{displaymath}$

Carl David
1999-01-29

What is K?

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Last edit: 13hrs:32min, Friday, January 29, 1999